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Last time, we saw a specific zero-knowledge proof for graph isomorphism. This introduced us to the concept of an interactive proof, where you have a prover and a verifier sending messages back and forth, and the prover is trying to prove a specific claim to the verifier.

A zero-knowledge proof is a special kind of interactive proof in which the prover has some secret piece of knowledge that makes it very easy to verify a disputed claim is true. The prover’s goal, then, is to convince the verifier (a polynomial-time algorithm) that the claim is true without revealing any knowledge at all about the secret.

In this post we’ll see that, using a bit of cryptography, zero-knowledge proofs capture a much wider class of problems than graph isomorphism. Basically, if you believe that cryptography exists, every problem whose answers can be easily verified have zero-knowledge proofs (i.e., all of the class NP). Here are a bunch of examples. For each I’ll phrase the problem as a question, and then say what sort of data the prover’s secret could be.

Given a boolean formula, is there an assignment of variables making it true? Secret: a satisfying assignment to the variables.

Given a set of integers, is there a subset whose sum is zero? Secret: such a subset.

Given a graph, does it have a 3-coloring? Secret: a valid 3-coloring.

Given a boolean circuit, can it produce a specific output? Secret: a choice of inputs that produces the output.

The common link among all of these problems is that they are NP-hard (graph isomorphism isn’t known to be NP-hard). For us this means two things: (1) we think these problems are actually hard, so the verifier can’t solve them, and (2) if you show that one of them has a zero-knowledge proof, then they all have zero-knowledge proofs.

We’re going to describe and implement a zero-knowledge proof for graph 3-colorability, and in the next post we’ll dive into the theoretical definitions and talk about the proof that the scheme we present is zero-knowledge. As usual, all of the code used in making this post is available in a repository on this blog’s Github page. In the follow up to this post, we’ll dive into more nitty gritty details about the proof that this works, and study different kinds of zero-knowledge.

One-way permutations

In a recent program gallery post we introduced the Blum-Blum-Shub pseudorandom generator. A pseudorandom generator is simply an algorithm that takes as input a short random string of length and produces as output a longer string, say, of length . This output string should not be random, but rather “indistinguishable” from random in a sense we’ll make clear next time. The underlying function for this generator is the “modular squaring” function , for some cleverly chosen . The is chosen in such a way that makes this mapping a permutation. So this function is more than just a pseudorandom generator, it’s a one-way permutation.

If you have a primality-checking algorithm on hand (we do), then preparing the Blum-Blum-Shub algorithm is only about 15 lines of code.

The interested reader should check out the proof gallery post for more details about this generator. For us, having a one-way permutation is the important part (and we’re going to defer the formal definition of “one-way” until next time, just think “hard to get inputs from outputs”).

The other concept we need, which is related to a one-way permutation, is the notion of a hardcore predicate. Let be a one-way permutation, and let be a function that produces a single bit from a string. We say that is a hardcore predicate for if you can’t reliably compute when given only .

Hardcore predicates are important because there are many one-way functions for which, when given the output, you can guess part of the input very reliably, but not the rest (e.g., if is a one-way function, is also one-way, but the part is trivially guessable). So a hardcore predicate formally measures, when given the output of a one-way function, what information derived from the input is hard to compute.

In the case of Blum-Blum-Shub, one hardcore predicate is simply the parity of the input bits.

def parity(n):
return sum(int(x) for x in bin(n)[2:]) % 2

Bit Commitment Schemes

A core idea that will makes zero-knowledge proofs work for NP is the ability for the prover to publicly “commit” to a choice, and later reveal that choice in a way that makes it infeasible to fake their commitment. This will involve not just the commitment to a single bit of information, but also the transmission of auxiliary data that is provably infeasible to fake.

Our pair of one-way permutation and hardcore predicate comes in very handy. Let’s say I want to commit to a bit . Let’s fix a security parameter that will measure how hard it is to change my commitment post-hoc, say . My process for committing is to draw a random string of length , and send you the pair , where is the XOR operator on two bits.

The guarantee of a one-way permutation with a hardcore predicate is that if you only see , you can’t guess with any reasonable edge over random guessing. Moreover, if you fix a bit , and take an unpredictably random bit , the XOR is also unpredictably random. In other words, if is hardcore, then so is for a fixed bit . Finally, to reveal my commitment, I just send the string and let you independently compute . Since is a permutation, that is the only that could have produced the commitment I sent you earlier.

Here’s a Python implementation of this scheme. We start with a generic base class for a commitment scheme.

Note that the “reveal” step is always simply to reveal the secret. Here’s the implementation subclass. We should also note that the security string should be chosen at random anew for every bit you wish to commit to. In this post we won’t reuse CommitmentScheme objects anyway.

One important detail is that the Blum-Blum-Shub one-way permutation is only a permutation when restricted to quadratic residues. As such, we generate our secret by shooting a random string through the one-way permutation to get a random residue. In fact this produces a uniform random residue, since the Blum-Blum-Shub modulus is chosen in such a way that ensures every residue has exactly four square roots.

As an exercise, write a program to verify that no other input to the Blum-Blum-Shub one-way permutation gives a valid verification. Test it on a small security parameter like .

It’s also important to point out that the verifier needs to do some additional validation that we left out. For example, how does the verifier know that the revealed secret actually is a quadratic residue? In fact, detecting quadratic residues is believed to be hard! To get around this, we could change the commitment scheme reveal step to reveal the random string that was used as input to the permutation to get the residue (cf. BBSCommitmentScheme.generateSecret for the random string that needs to be saved/revealed). Then the verifier could generate the residue in the same way. As an exercise, upgrade the bit commitment an verifier classes to reflect this.

In order to get a zero-knowledge proof for 3-coloring, we need to be able to commit to one of three colors, which requires two bits. So let’s go overkill and write a generic integer commitment scheme. It’s simple enough: specify a bound on the size of the integers, and then do an independent bit commitment for every bit.

Before we move on, we should note that this integer commitment scheme “blows up” the secret by quite a bit. If you have a security parameter and an integer with bits, then the commitment uses roughly bits. A more efficient method would be to simply use a good public-key encryption scheme, and then reveal the secret key used to encrypt the message. While we implemented such schemes previously on this blog, I thought it would be more fun to do something new.

A zero-knowledge proof for 3-coloring

First, a high-level description of the protocol. The setup: the prover has a graph with vertices and edges , and also has a secret 3-coloring of the vertices . Recall, a 3-coloring is just an assignment of colors to vertices (in this case the colors are 0,1,2) so that no two adjacent vertices have the same color.

So the prover has a coloring to be kept secret, but wants to prove that is 3-colorable. The idea is for the verifier to pick a random edge , and have the prover reveal the colors of and . However, if we run this protocol only once, there’s nothing to stop the prover from just lying and picking two distinct colors. If we allow the verifier to run the protocol many times, and the prover actually reveals the colors from their secret coloring, then after roughly rounds the verifier will know the entire coloring. Each step reveals more knowledge.

We can fix this with two modifications.

The prover first publicly commits to the coloring using a commitment scheme. Then when the verifier asks for the colors of the two vertices of a random edge, he can rest assured that the prover fixed a coloring that does not depend on the verifier’s choice of edge.

The prover doesn’t reveal colors from their secret coloring, but rather from a random permutation of the secret coloring. This way, when the verifier sees colors, they’re equally likely to see any two colors, and all the verifier will know is that those two colors are different.

So the scheme is: prover commits to a random permutation of the true coloring and sends it to the verifier; the verifier asks for the true colors of a given edge; the prover provides those colors and the secrets to their commitment scheme so the verifier can check.

The key point is that now the verifier has to commit to a coloring, and if the coloring isn’t a proper 3-coloring the verifier has a reasonable chance of picking an improperly colored edge (a one-in- chance, which is at least ). On the other hand, if the coloring is proper, then the verifier will always query a properly colored edge, and it’s zero-knowledge because the verifier is equally likely to see every pair of colors. So the verifier will always accept, but won’t know anything more than that the edge it chose is properly colored. Repeating this -ish times, with high probability it’ll have queried every edge and be certain the coloring is legitimate.

Let’s implement this scheme. First the data types. As in the previous post, graphs are represented by edge lists, and a coloring is represented by a dictionary mapping a vertex to 0, 1, or 2 (the “colors”).

In commitToColoring we randomly permute the underlying colors, and then compose that permutation with the secret coloring, committing to each resulting color independently. In revealColors we reveal only those colors for a queried edge. Note that we don’t actually need to store the permuted coloring, because it’s implicitly stored in the commitments.

It’s crucial that we reject any query that doesn’t correspond to an edge. If we don’t reject such queries then the verifier can break the protocol! In particular, by querying non-edges you can determine which pairs of nodes have the same color in the secret coloring. You can then chain these together to partition the nodes into color classes, and so color the graph. (After seeing the Verifier class below, implement this attack as an exercise).

So while we haven’t proved it rigorously, we’ve seen the zero-knowledge proof for graph 3-coloring. This automatically gives us a zero-knowledge proof for all of NP, because given any NP problem you can just convert it to the equivalent 3-coloring problem and solve that. Of course, the blowup required to convert a random NP problem to 3-coloring can be polynomially large, which makes it unsuitable for practice. But the point is that this gives us a theoretical justification for which problems have zero-knowledge proofs in principle. Now that we’ve established that you can go about trying to find the most efficient protocol for your favorite problem.

Anticipatory notes

When we covered graph isomorphism last time, we said that a simulator could, without participating in the zero-knowledge protocol or knowing the secret isomorphism, produce a transcript that was drawn from the same distribution of messages as the protocol produced. That was all that it needed to be “zero-knowledge,” because anything the verifier could do with its protocol transcript, the simulator could do too.

We can do exactly the same thing for 3-coloring, exploiting the same “reverse order” trick where the simulator picks the random edge first, then chooses the color commitment post-hoc.

Unfortunately, both there and here I’m short-changing you, dear reader. The elephant in the room is that our naive simulator assumes the verifier is playing by the rules! If you want to define security, you have to define it against a verifier who breaks the protocol in an arbitrary way. For example, the simulator should be able to produce an equivalent transcript even if the verifier deterministically picks an edge, or tries to pick a non-edge, or tries to send gibberish. It takes a lot more work to prove security against an arbitrary verifier, but the basic setup is that the simulator can no longer make choices for the verifier, but rather has to invoke the verifier subroutine as a black box. (To compensate, the requirements on the simulator are relaxed quite a bit; more on that next time)

Because an implementation of such a scheme would involve a lot of validation, we’re going to defer the discussion to next time. We also need to be more specific about the different kinds of zero-knowledge, since we won’t be able to achieve perfect zero-knowledge with the simulator drawing from an identical distribution, but rather a computationally indistinguishable distribution.

We’ll define all this rigorously next time, and discuss the known theoretical implications and limitations. Next time will be cuffs-off theory, baby!

An integer is called a quadratic residue of another integer if it can be written as for some . That is, if it’s the remainder when dividing a perfect square by . Some numbers, like , have very special patterns in their quadratic residues, only 0, 1, and 4 can occur as quadratic residues.

The core idea behind this random number generator is that, for a specially chosen modulus , telling whether a number is a quadratic residue mod is hard. In fact, one can directly convert an algorithm that can predict the next bit of this random number generator (by even a slight edge) into an arbitrarily accurate quadratic-residue-decider. So if computing quadratic residues is even mildly hard, then predicting the next bit in this random number generator is very hard.

More specifically, the conjectured guarantee about this random number generator is the following: if you present a polynomial time adversary with two sequences:

A truly random sequence of bits of length ,

bits from the output of the pseudorandom generator when seeded with a starting state shorter than bits.

Then the adversary can’t distinguish between the two sequences with probability “significantly” more than 1/2, where by “significantly” I mean for any (i.e., the edge over randomness vanishes faster than any inverse polynomial). It turns out, due to a theorem of Yao, that this is equivalent to not being able to guess the next bit in a pseudorandom sequence with a significant edge over a random guess, even when given the previous bits in the sequence (or any bits in the sequence).

This emphasizes a deep philosophical viewpoint in theoretical computer science, that whether some object has a property (randomness) really only depends on the power of a computationally limited observer to identify that property. If nobody can tell the difference between fake randomness and real randomness, then the fake randomness is random. Offhand I wonder whether you can meaningfully apply this view to less mathematical concepts like happiness and status.

Anyway, the modulus is chosen in such a way that every quadratic residue of has a unique square root which is also a quadratic residue. This makes the squaring function a bijection on quadratic residues. In other words, with a suitably chosen , there’s no chance that we’ll end up with where there are very few quadratic residues and the numbers output by the Blum-Blum-Shub generator have a short cycle. Moreover, the assumption that detecting quadratic residues mod is hard makes the squaring function a one-way permutation.

Like this:

This post is intended for people with a little bit of programming experience and no prior mathematical background.

So let’s talk about numbers.

Numbers are curious things. On one hand, they represent one of the most natural things known to humans, which is quantity. It’s so natural to humans that even newborn babies are in tune with the difference between quantities of objects between 1 and 3, in that they notice when quantity changes much more vividly than other features like color or shape.

But our familiarity with quantity doesn’t change the fact that numbers themselves (as an idea) are a human invention. And they’re not like most human inventions, the kinds where you have to tinker with gears or circuits to get a machine that makes your cappuccino. No, these are mathematical inventions. These inventions exist only in our minds.

Numbers didn’t always exist. A long time ago, back when the Greeks philosophers were doing their philosophizing, negative numbers didn’t exist! In fact, it wasn’t until 1200 AD that the number zero was first considered in Europe. Zero, along with negative numbers and fractions and square roots and all the rest, were invented primarily to help people solve more problemsthan they could with the numbers they had available. That is, numbers were invented primarily as a way for people to describe their ideas in a useful way. People simply wondered “is there a number whose square gives you 2?” And after a while they just decided there was and called it because they didn’t have a better name for it.

But with these new solutions came a host of new problems. You see, although I said mathematical inventions only exist in our minds, once they’re invented they gain a life of their own. You start to notice patterns in your mathematical objects and you have to figure out why they do the things they do. And numbers are a perfectly good example of this: once I notice that I can multiply a number by itself, I can ask how often these “perfect squares” occur. That is, what’s the pattern in the numbers ? If you think about it for a while, you’ll find that square numbers have a very special relationship with odd numbers.

Other times, however, the things you invent turn out to make no sense at all, and you can prove they never existed in the first place! It’s an odd state of affairs, but we’re going to approach the subject of complex numbers from this mindset. We’re going to come up with a simple idea, the idea that negative numbers can be perfect squares, and explore the world of patterns it opens up. Along the way we’ll do a little bit of programming to help explore, give some simple proofs to solidify our intuition, and by the end we’ll see how these ideas can cause wonderful patterns like this one:

The number i

Let’s bring the story back around to squares. One fact we all remember about numbers is that squaring a number gives you something non-negative. , and so on. But it certainly doesn’t have to be this way. What if we got sick of that stupid fact and decided to invent a new number whose square was negative? Which negative, you ask? Well it doesn’t really matter, because I can always stretch it larger or smaller so that it’s square is -1.

Let’s see how: if you say that your made-up number makes , then I can just use to get a number whose square is -1. If you’re going to invent a number that’s supposed to interact with our usual numbers, then you have to be allowed to add, subtract, and multiply with regular old real numbers, and the usual properties would have to still work. So it would have to be true that .

So because it makes no difference (this is what mathematicians mean by, “without loss of generality”) we can assume that the number we’re inventing will have a square of negative one. Just to line up with history, let’s call the new number . So there it is: exists and . And now that we are “asserting” that plays nicely with real numbers, we get these natural rules for adding and subtracting and multiplying and dividing. For example

is a new number, which we’ll just call . And if we added two of these together, , we can combine the real parts and the parts to get . Same goes for subtraction. In general a complex number looks like , because as we’ll see in the other points you can simplify every simple arithmetic expression down to just one “real number” part and one “real number times ” part.

We can multiply , and we’ll just call it , and we require that multiplication distributes across addition (that the FOIL rule works). So that, for example, .

Dividing is a significantly more annoying. Say we want to figure out what is (in fact, it’s not even obvious that this should look like a regular number! But it does). The notation just means we’re looking for a number which, when we multiply by the denominator , we get back to 1. So we’re looking to find out when where and are variables we’re trying to solve for. If we multiply it out we get , and since the real part and the part have to match up, we know that and . If we solve these two equations, we find that works great. If we want to figure out something like , we just find out what is first, and then multiply the result by .

So that was tedious and extremely boring, and we imagine you didn’t even read it (that’s okay, it really is boring!). All we’re doing is establishing ground rules for the game, so if you come across some arithmetic that doesn’t make sense, you can refer back to this list to see what’s going on. And once again, for the purpose of this post, we’re asserting that all these laws hold. Maybe some laws follow from others, but as long as we don’t come up with any nasty self-contradictions we’ll be fine.

And now we turn to the real questions: is the only square root of -1? Does itself have a square root? If it didn’t, we’d be back to where we started, with some numbers (the non- numbers) having square roots while others don’t. And so we’d feel the need to make all the numbers happy by making up more numbers to be their square roots, and then worrying what if these new numbers don’t have square roots and…gah!

I’ll just let you in on the secret to save us from this crisis. It turns out that does have a square root in terms of other numbers, but in order to find it we’ll need to understand from a different angle, and that angle turns out to be geometry.

Geometry? How is geometry going to help me understand numbers!?

It’s a valid question and part of why complex numbers are so fascinating. And I don’t mean geometry like triangles and circles and parallel lines (though there will be much talk of angles), I mean transformations in the sense that we’ll be “stretching,” “squishing,” and “rotating” numbers. Maybe another time I can tell you why for me “geometry” means stretching and rotating; it’s a long but very fun story.

The clever insight is that you can represent complex numbers as geometric objects in the first place. To do it, you just think of as a pair of numbers , (the pair of real part and part), and then plot that point on a plane. For us, the -axis will be the “real” axis, and the -axis will be the -axis. So the number is plotted 3 units in the positive direction and 4 units in the negative direction. Like this:

The “j” instead of “i” is not a typo, but a disappointing fact about the programming language we used to make this image. We’ll talk more about why later.

We draw it as an arrow for a good reason. Stretching, squishing, rotating, and reflecting will all be applied to the arrow, keeping its tail fixed at the center of the axes. Sometimes the arrow is called a “vector,” but we won’t use that word because here it’s synonymous with “complex number.”

So let’s get started squishing stuff.

Stretching, Squishing, Rotating

Before we continue I should clear up some names. We call a number that has an in it a complex number, and we call the part without the the real part (like 2 in ) and the part with the complex part.

Python is going to be a great asset for us in exploring complex numbers, so let’s jump right into it. It turns out that Python natively supports complex numbers, and I wrote a program for drawing complex numbers. I used it to make the plot above. The program depends on a library I hate called matplotlib, and so the point of the program is to shield you from as much pain as possible and focus on complex numbers. You can use the program by downloading it from this blog’s Github page, along with everything else I made in writing this post. All you need to know how to do is call a function, and I’ve done a bit of window dressing removal to simplify things (I really hate matplotlib).

Before we show some examples of how to use it, we have to understand how to use complex numbers in Python. It’s pretty simple, except that Python was written by people who hate math, and so they decided the complex number would be represented by instead of (people who hate math are sometimes called “engineers,” and they use out of spite. Not really, though).

So in Python it’s just like any other computation. For example:

>>> (1 + 1j)*(4 - 2j) == (6+2j)
True
>>> 1 / (1+1j)
(0.5-0.5j)

And so calling the plotting function with a given list of complex numbers is as simple as importing the module and calling the function

So let’s use plots like this one to explore what “multiplication by ” does to a complex number. It might not seem exciting at first, but I promise there’s a neat punchline.

Even without plotting it’s pretty easy to tell what multiplying by does to some numbers. It takes 1 to , moves to , it takes -1 to , and to .

What’s the pattern in these? well if we plot all these numbers, they’re all at right angles in counter-clockwise order. So this might suggest that multiplication by does some kind of rotation. Is that always the case? Well lets try it with some other more complicated numbers. Click the plots below to enlarge.

Well, it looks close but it’s hard to tell. Some of the axes are squished and stretched, so it might be that our images don’t accurately represent the numbers (the real world can be such a pain). Well when visual techniques fail, we can attempt to prove it.

Clearly multiplying by does some kind of rotation, maybe with other stuff too, and it shouldn’t be so hard to see that multiplying by does the same thing no matter which number you use (okay, the skeptical readers will say that’s totally hard to see, but we’ll prove it super rigorously in a minute). So if we take any number and multiply it by once, then twice, then three times, then four, and if we only get back to where we started at four multiplications, then each rotation had to be a quarter turn.

Indeed,

This still isn’t all that convincing, and we want to be 100% sure we’re right. What we really need is a way to arithmetically compute the angle between two complex numbers in their plotted forms. What we’ll do is find a way to measure the angle of one complex number with the -axis, and then by subtraction we can get angles between arbitrary points. For example, in the figure below .

One way to do this is with trigonometry: the geometric drawing of is the hypotenuse of a right triangle with the -axis.

And so if is the length of the arrow, then by the definition of sine and cosine, . If we have , and , we can solve for a unique and , so instead of representing a complex number in terms of the pair of numbers , we can represent it with the pair of numbers . And the conversion between the two is just

The representation is called the polar representation, while the representation is called the rectangular representation or the Cartesian representation. Converting between polar and Cartesian coordinates fills the pages of many awful pre-calculus textbooks (despite the fact that complex numbers don’t exist in classical calculus). Luckily for us Python has built-in functions to convert between the two representations for us.

It’s a little bit inaccurate on the rounding, but it’s fine for our purposes.

So how do we compute the angle between two complex numbers? Just convert each to the polar form, and subtract the second coordinates. So if we get back to our true goal, to figure out what multiplication by does, we can just do everything in polar form. Here’s a program that computes the angle between two complex numbers.

Note that the decimal form of is 1.57079…, and that the negative angle is equivalent to if you add a full turn of to it. So programmatically we can see that for every input we try multiplying by rotates 90 degrees.

But we still haven’t proved it works. So let’s do that now. To say what the angle is between and , we need to transform the second number into the usual polar form (where the is on the sine part and not the cosine part). But we know, or I’m telling you now, this nice fact about sine and cosine:

This fact is maybe awkward to write out algebraically, but it’s just saying that if you shift the whole sine curve a little bit you get the cosine curve, and if you keep shifting it you get the opposite of the sine curve (and if you kept shifting it even more you’d eventually get back to the sine curve; they’re called periodic for this reason).

So immediately we can rewrite the second number as . The angle is the same as the original angle plus a right angle of . Neat!

Applying this same idea to , it’s not much harder to prove that multiplying two complex numbers in general multiplies their lengths and adds their angles. So if a complex number has its magnitude smaller than 1, multiplying by squishes and rotates whatever is being multiplied. And if the magnitude is greater than 1, it stretches and rotates. So we have a super simple geometric understanding of how arithmetic with complex numbers works. And as we’re about to see, all this stretching and rotating results in some really weird (and beautifully mysterious!) mathematics and programs.

But before we do that we still have one question to address, the question that started this whole geometric train of thought: does have a square root? Indeed, I’m just looking for a number such that, when I square its length and double its angle, I get . Indeed, the angle we want is , and the length we want is , which means . Sweet! There is another root if you play with the signs, see if you can figure it out.

In fact it’s a very deeper and more beautiful theorem (“theorem” means “really important fact”) called the fundamental theorem of algebra. And essentially it says that the complex numbers are complete. That is, we can always find square roots, cube roots, or anything roots of numbers involving . It actually says a lot more, but it’s easier to appreciate the rest of it after you do more math than we’re going to do in this post.

On to pretty patterns!

The Fractal

So here’s a little experiment. Since every point in the plane is the end of some arrow representing a complex number, we can imagine transforming the entire complex plane by transforming each number by the same rule. The most interesting simple rule we can think of: squaring! So though it might strain your capacity for imagination, try to visualize the idea like this. Squaring a complex number is the same as squaring it’s length and doubling its angle. So imagine: any numbers whose arrows are longer than 1 will grow much bigger, arrows shorter than 1 will shrink, and arrows of length exactly one will stay the same length (arrows close to length 1 will grow/shrink much more slowly than those far away from 1). And complex numbers with small positive angles will increase their angle, but only a bit, while larger angles will grow faster.

Here’s an animation made by Douglas Arnold showing what happens to the set of complex numbers with or . Again, imagine every point is the end of a different arrow for the corresponding complex number. The animation is for a single squaring, and the points move along the arc they would travel if one rotated/stretched them smoothly.

So that’s pretty, but this is by all accounts a well-behaved transformation. It’s “predictable,” because for example we can always tell which complex numbers will get bigger and bigger (in length) and which will get smaller.

What if, just for the sake of tinkering, we changed the transformation a little bit? That is, instead of sending to (I’ll often write this ), what if we sent

Now it’s not so obvious: which numbers will grow and which will shrink? Notice that it’s odd because adding 1 only changes the real part of the number. So a number whose length is greater than 1 can become small under this transformation. For example, is sent to , so something slightly larger would also be close to zero. Indeed, .

So here’s an interesting question: are there any complex numbers that will stay small even if I keep transforming like this forever? Specifically, if I call , and I call , and likewise call for repeated transformations of , is there a number so that for every, the value ? “Obvious” choices like don’t work, and neither do random guesses like or . So should we guess the answer is no?

Before we jump to conclusions let’s write a program to see what happens for more than our random guesses. The program is simple: we’ll define the “square plus one” function, and then repeatedly apply that function to a number for some long number of times (say, 250 times). If the length of the number stays under 2 after so many tries, we’ll call it “small forever,” and otherwise we’ll call it “not small forever.”

This isSmallForever function is generic: you can give it any function and it will repeatedly call on until the result grows bigger than 2 in length. Note that the abs function is a built-in Python function for computing the length of a complex number.

Then I wrote a classify function, which you can give a window and a small increment, and it will produce a grid of zeros and ones marking the results of isSmallForever. The details of the function are not that important. I also wrote a function that turns the grid into a picture. So here’s an example of how we’d use it:

And here’s the result. Points colored black grow beyond 2, and white points stay small for the whole test.

Looks like they’ll always grow big.

So it looks like repeated squaring plus one will always make complex numbers grow big. That’s not too exciting, but we can always make it more exciting. What happens if we replace the 1 in with a different complex number? For example, if we do then will things always grow big?

You can randomly guess and see that 0 will never grow big, because and . It will just oscillate forever. So with -1 some numbers will grow and some will not! Let’s use the same routine above to see which:

Gorgeous. If you try this at home you’ll notice, however, that this took a hell of a long time to run. Speeding up our programs is very possible, but it’s a long story for another time. For now we can just be patient.

Indeed, this image has a ton of interesting details! It looks almost circular in the middle, but if we zoom in we can see that it’s more like a rippling wave

It’s pretty incredible, and a huge question is jumping out at me: what the heck is causing this pattern to occur? What secret does -1 know that +1 doesn’t that makes the resulting pattern so intricate?

But an even bigger question is this. We just discovered that some values of make result in interesting patterns, and some do not! So the question is which ones make interesting patterns? Even if we just, say, fix the starting point to zero: what is the pattern in the complex numbers that would tell me when this transformation makes zero blow up, and when it keeps zero small?

Sounds like a job for another program. This time we’ll use a nice little Python feature called a closure, which we define a function that saves the information that exists when it’s created for later. It will let us write a function that takes in and produces a function that transforms according to .

def squarePlusC(c):
def f(z):
return z*z + c
return f

And we can use the very same classification/graphing function from before to do this.

Stunning. This wonderful pattern, which is still largely not understood today, is known as the Mandelbrot set. That is, the white points are the points in the Mandlebrot set, and the black points are not in it. The detail on the border of this thing is infinitely intricate. For example, we can change the window in our little program to zoom in on a particular region.

And if you keep zooming in you keep getting more and more detail. This was true of the specific case of , but somehow the patterns in the Mandelbrot set are much more varied and interesting. And if you keep going down eventually you’ll see patterns that look like the original Mandelbrot set. We can already kind of see that happening above. The name for this idea is a fractal, and the image has it too. Fractals are a fascinating and mysterious subject studied in a field called discrete dynamical systems. Many people dedicate their entire lives to studying these things, and it’s for good reason. There’s a lot to learn and even more that’s unknown!

So this is the end of our journey for now. I’ve posted all of the code we used in the making of this post so you can continue to play, but here are some interesting ideas.

The Mandelbrot set (and most fractals) are usually colored. The way they’re colored is as follows. Rather than just say true or false when zero blows up beyond 2 in length, you return the number of iterations that happened. Then you pick a color based on how big is. There’s a link below that lets you play with this. In fact, adding colors shows that there is even more intricate detail happening outside the Mandelbrot set that’s too faint to see in our pictures above. Such as this.

Some very simple questions about fractals are very hard to answer. For example, is the Mandelbrot set connected? That is, is it possible to “walk” from every point in the Mandelbrot set to every other point without leaving the set? Despite the scattering of points in the zoomed in picture above that suggest the answer is no, the answer is actually yes! This is a really difficult thing to prove, however.

You should definitely be experimenting with this stuff! What happens if you change the length threshold from 2 to some bigger number? What about a smaller number? What if you do powers different than ? There’s so much to explore!

The big picture thing to take away from this is that it’s not the numbers themselves that are particularly interesting, it’s the transformations of the numbers that generate these patterns! The interesting questions are what kinds of things are the same under these transformations, and what things are different. This is a very general idea in mathematics, and the more math you do the more you’ll find yourself wondering about useful and bizarre transformations.

History

As the reader has heard countless times in this series, an elliptic curve is a geometric object whose points have a surprising and well-defined notion of addition. That you can add some points on some elliptic curves was a well-known technique since antiquity, discovered by Diophantus. It was not until the mid 19th century that the general question of whether addition always makes sense was answered by Karl Weierstrass. In 1908 Henri Poincaré asked about how one might go about classifying the structure of elliptic curves, and it was not until 1922 that Louis Mordell proved the fundamental theorem of elliptic curves, classifying their algebraic structure for most important fields.

While mathematicians have always been interested in elliptic curves (there is currently a million dollar prize out for a solution to one problem about them), its use in cryptography was not suggested until 1985. Two prominent researchers independently proposed it: Neal Koblitz at the University of Washington, and Victor Miller who was at IBM Research at the time. Their proposal was solid from the start, but elliptic curves didn’t gain traction in practice until around 2005. More recently, the NSA was revealed to have planted vulnerable national standards for elliptic curve cryptography so they could have backdoor access. You can see a proof and implementation of the backdoor at Aris Adamantiadis’s blog. For now we’ll focus on the cryptographic protocols themselves.

The Discrete Logarithm Problem

Koblitz and Miller had insights aplenty, but the central observation in all of this is the following.

Adding is easy on elliptic curves, but undoing addition seems hard.

What I mean by this is usually called the discrete logarithm problem. Here’s a formal definition. Recall that an additive group is just a set of things that have a well-defined addition operation, and the that notation means ( times).

Definition: Let be an additive group, and let be elements of so that for some integer . The discrete logarithm problem asks one to find when given and .

I like to give super formal definitions first, so let’s do a comparison. For integers this problem is very easy. If you give me 12 and 4185072, I can take a few seconds and compute that using the elementary-school division algorithm (in the above notation, , and ). The division algorithm for integers is efficient, and so it gives us a nice solution to the discrete logarithm problem for the additive group of integers .

The reason we use the word “logarithm” is because if your group operation is multiplication instead of addition, you’re tasked with solving the equation for . With real numbers you’d take a logarithm of both sides, hence the name. Just in case you were wondering, we can also solve the multiplicative logarithm problem efficiently for rational numbers (and hence for integers) using the square-and-multiply algorithm. Just square until doing so would make you bigger than , then multiply by until you hit .

But integers are way nicer than they need to be. They are selflessly well-ordered. They give us division for free. It’s a computational charity! What happens when we move to settings where we don’t have a division algorithm? In mathematical lingo: we’re really interested in the case when is just a group, and doesn’t have additional structure. The less structure we have, the harder it should be to solve problems like the discrete logarithm. Elliptic curves are an excellent example of such a group. There is no sensible ordering for points on an elliptic curve, and we don’t know how to do division efficiently. The best we can do is add to itself over and over until we hit , and it could easily happen that (as a number) is exponentially larger than the number of bits in and .

What we really want is a polynomial time algorithm for solving discrete logarithms. Since we can take multiples of a point very fast using the double-and-add algorithm from our previous post, if there is no polynomial time algorithm for the discrete logarithm problem then “taking multiples” fills the role of a theoretical one-way function, and as we’ll see this opens the door for secure communication.

Here’s the formal statement of the discrete logarithm problem for elliptic curves.

Problem: Let be an elliptic curve over a finite field . Let be points on such that for some integer . Let denote the number of bits needed to describe the point . We wish to find an algorithm which determines and has runtime polynomial in . If we want to allow randomness, we can require the algorithm to find the correct with probability at least 2/3.

So this problem seems hard. And when mathematicians and computer scientists try to solve a problem for many years and they can’t, the cryptographers get excited. They start to wonder: under the assumption that the problem has no efficient solution, can we use that as the foundation for a secure communication protocol?

The Diffie-Hellman Protocol and Problem

Let’s spend the rest of this post on the simplest example of a cryptographic protocol based on elliptic curves: the Diffie-Hellman key exchange.

A lot of cryptographic techniques are based on two individuals sharing a secret string, and using that string as the key to encrypt and decrypt their messages. In fact, if you have enough secret shared information, and you only use it once, you can have provably unbreakable encryption! We’ll cover this idea in a future series on the theory of cryptography (it’s called a one-time pad, and it’s not all that complicated). All we need now is motivation to get a shared secret.

Because what if your two individuals have never met before and they want to generate such a shared secret? Worse, what if their only method of communication is being monitored by nefarious foes? Can they possibly exchange public information and use it to construct a shared piece of secret information? Miraculously, the answer is yes, and one way to do it is with the Diffie-Hellman protocol. Rather than explain it abstractly let’s just jump right in and implement it with elliptic curves.

As hinted by the discrete logarithm problem, we only really have one tool here: taking multiples of a point. So say we’ve chosen a curve and a point on that curve . Then we can take some secret integer , and publish and for the world to see. If the discrete logarithm problem is truly hard, then we can rest assured that nobody will be able to discover .

How can we use this to established a shared secret? This is where Diffie-Hellman comes in. Take our two would-be communicators, Alice and Bob. Alice and Bob each pick a binary string called a secret key, which in interpreted as a number in this protocol. Let’s callAlice’s secret key and Bob’s , and note that they don’t have to be the same. As the name “secret key” suggests, the secret keys are held secret. Moreover, we’ll assume that everything else in this protocol, including all data sent between the two parties, is public.

So Alice and Bob agree ahead of time on a public elliptic curve and a public point on . We’ll sometimes call this point the base point for the protocol.

Bob can cunningly do the following trick: take his secret key and send to Alice. Equally slick Alice computes and sends that to Bob. Now Alice, having , computes . And Bob, since he has , can compute . But since addition is commutative in elliptic curve groups, we know . The secret piece of shared information can be anything derived from this new point, for example its -coordinate.

If we want to talk about security, we have to describe what is public and what the attacker is trying to determine. In this case the public information consists of the points . What is the attacker trying to figure out? Well she really wants to eavesdrop on their subsequent conversation, that is, the stuff that encrypt with their new shared secret . So the attacker wants find out . And we’ll call this the Diffie-Hellman problem.

Diffie-Hellman Problem: Suppose you fix an elliptic curve over a finite field , and you’re given four points and for some unknown integers . Determine if in polynomial time (in the lengths of ).

On one hand, if we had an efficient solution to the discrete logarithm problem, we could easily use that to solve the Diffie-Hellman problem because we could compute and them quickly compute and check if it’s . In other words discrete log is at least as hard as this problem. On the other hand nobody knows if you can do this without solving the discrete logarithm problem. Moreover, we’re making this problem as easy as we reasonably can because we don’t require you to be able to compute . Even if some prankster gave you a candidate for , all you have to do is check if it’s correct. One could imagine some test that rules out all fakes but still doesn’t allow us to compute the true point, which would be one way to solve this problem without being able to solve discrete log.

So this is our hardness assumption: assuming this problem has no efficient solution then no attacker, even with really lucky guesses, can feasibly determine Alice and Bob’s shared secret.

Python Implementation

The Diffie-Hellman protocol is just as easy to implement as you would expect. Here’s some Python code that does the trick. Note that all the code produced in the making of this post is available on this blog’s Github page.

Pythons os module allows us to access the operating system’s random number generator (which is supposed to be cryptographically secure) via the function urandom, which accepts as input the number of bytes you wish to generate, and produces as output a Python bytestring object that we then convert to an integer. Our simplistic (and totally insecure!) protocol uses the elliptic curve defined by over the finite field . We pick the base point , and call the relevant functions with placeholders for actual network transmission functions.

There is one issue we have to note. Say we fix our base point . Since an elliptic curve over a finite field can only have finitely many points (since the field only has finitely many possible pairs of numbers), it will eventually happen that is the ideal point. Recall that the smallest value of for which is called the order of . And so when we’re generating secret keys, we have to pick them to be smaller than the order of the base point. Viewed from the other angle, we want to pick to have large order, so that we can pick large and difficult-to-guess secret keys. In fact, no matter what integer you use for the secret key it will be equivalent to some secret key that’s less than the order of . So if an attacker could guess the smaller secret key he wouldn’t need to know your larger key.

The base point we picked in the example above happens to have order 1964, so an 8-bit key is well within the bounds. A real industry-strength elliptic curve (say, Curve25519 or the curves used in the NIST standards*) is designed to avoid these problems. The order of the base point used in the Diffie-Hellman protocol for Curve25519 has gargantuan order (like ). So 256-bit keys can easily be used. I’m brushing some important details under the rug, because the key as an actual string is derived from 256 pseudorandom bits in a highly nontrivial way.

So there we have it: a simple cryptographic protocol based on elliptic curves. While we didn’t experiment with a truly secure elliptic curve in this example, we’ll eventually extend our work to include Curve25519. But before we do that we want to explore some of the other algorithms based on elliptic curves, including random number generation and factoring.

Comments on Insecurity

Why do we use elliptic curves for this? Why not do something like RSA and do multiplication (and exponentiation) modulo some large prime?

Well, it turns out that algorithmic techniques are getting better and better at solving the discrete logarithm problem for integers mod , leading some to claim that RSA is dead. But even if we will never find a genuinely efficient algorithm (polynomial time is good, but might not be good enough), these techniques have made it clear that the key size required to maintain high security in RSA-type protocols needs to be really big. Like 4096 bits. But for elliptic curves we can get away with 256-bit keys. The reason for this is essentially mathematical: addition on elliptic curves is not as well understood as multiplication is for integers, and the more complex structure of the group makes it seem inherently more difficult. So until some powerful general attacks are found, it seems that we can get away with higher security on elliptic curves with smaller key sizes.

I mentioned that the particular elliptic curve we chose was insecure, and this raises the natural question: what makes an elliptic curve/field/basepoint combination secure or insecure? There are a few mathematical pitfalls (including certain attacks we won’t address), but one major non-mathematical problem is called a side-channel attack. A side channel attack against a cryptographic protocol is one that gains additional information about users’ secret information by monitoring side-effects of the physical implementation of the algorithm.

The problem is that different operations, doubling a point and adding two different points, have very different algorithms. As a result, they take different amounts of time to complete and they require differing amounts of power. Both of these can be used to reveal information about the secret keys. Despite the different algorithms for arithmetic on Weierstrass normal form curves, one can still implement them to be secure. Naively, one might pad the two subroutines with additional (useless) operations so that they have more similar time/power signatures, but I imagine there are better methods available.

But much of what makes a curve’s domain parameters mathematically secure or insecure is still unknown. There are a handful of known attacks against very specific families of parameters, and so cryptography experts simply avoid these as they are discovered. Here is a short list of pitfalls, and links to overviews:

Make sure the order of your basepoint has a short facorization (e.g., is or for some prime ). Otherwise you risk attacks based on the Chinese Remainder Theorem, the most prominent of which is called Pohlig-Hellman.

Make sure your curve is not supersingular. If it is you can reduce the discrete logarithm problem to one in a different and much simpler group.

If your curve is defined over , make sure the number of points on is not equal to . Such a curve is called prime-field anomalous, and its discrete logarithm problem can be reduced to the (additive) version on integers.

Don’t pick a small underlying field like for small . General-purpose attacks can be sped up significantly against such fields.

If you use the field , ensure that is prime. Many believe that if has small divisors, attacks based on some very complicated algebraic geometry can be used to solve the discrete logarithm problem more efficiently than any general-purpose method. This gives evidence that being composite at all is dangerous, so we might as well make it prime.

The interesting thing is that there is little about the algorithm and protocol that is vulnerable. Almost all of the vulnerabilities come from using bad curves, bad fields, or a bad basepoint. Since the known attacks work on a pretty small subset of parameters, one potentially secure technique is to just generate a random curve and a random point on that curve! But apparently all respected national agencies will refuse to call your algorithm “standards compliant” if you do this.

Next time we’ll continue implementing cryptographic protocols, including the more general public-key message sending and signing protocols.

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